CHAPTER 14: Confidence Intervals The Basics

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CHAPTER 14Confidence IntervalsThe Basics
ESSENTIAL STATISTICS Second Edition David S.
Moore, William I. Notz, and Michael A.
Fligner Lecture Presentation
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Chapter 14 Concepts

• The Reasoning of Statistical Estimation
• Margin of Error and Confidence Level
• Confidence Intervals for a Population Mean
• How Confidence Intervals Behave

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Chapter 14 Objectives

• Define statistical inference
• Describe the reasoning of statistical estimation
• Describe the parts of a confidence interval
• Interpret a confidence level
• Construct and interpret a confidence interval for
  the mean of a Normal population
• Describe how confidence intervals behave

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Statistical Inference
After we have selected a sample, we know the
responses of the individuals in the sample.
However, the reason for taking the sample is to
infer from that data some conclusion about the
wider population represented by the sample.
Statistical Inference Statistical inference
provides methods for drawing conclusions about a
population from sample data.
Population
Sample
Collect data from a representative Sample...
Make an Inference about the Population.
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Simple Conditions for InferenceAbout a Mean
This chapter presents the basic reasoning of
statistical inference. We start with a setting
that is too simple to be realistic.

• Simple Conditions for Inference About a Mean
• We have an SRS from the population of interest.
  There is no nonresponse or other practical
  difficulty.
• The variable we measure has an exactly Normal
  distribution N(µ,s) in the population.
• We dont know the population mean µ, but we do
  know the population standard deviation s.

Note The conditions that we have a perfect SRS,
that the population is exactly Normal, and that
we know the population standard deviation are all
unrealistic.
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The Reasoning of Statistical Estimation
Your instructor has selected a Mystery Mean
value µ and stored it as M in their calculator.
The following command was executed on their
calculator mean(randNorm(M,20,16))
The result was 240.79. This tells us the
calculator chose an SRS of 16 observations from a
Normal population with mean M and standard
deviation 20. The resulting sample mean of those
16 values was 240.79.
Suppose we want to determine an interval of
reasonable values for the population mean µ. We
can use the result above and what we learned
about sampling distributions in the previous
chapters.
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The Reasoning of Statistical Estimation
Since the sample mean is 240.79, we could guess
that µ is somewhere around 240.79. How close
to 240.79 is µ likely to be?
To answer this question, we must ask another
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The Reasoning of Statistical Estimation

• In repeated samples, the values of the sample
  mean will follow a Normal distribution with mean
  µ and standard deviation 5.

• The 68-95-99.7 Rule tells us that in 95 of all
  samples of size 16, the sample mean will be
  within 10 (two standard deviations) of µ.

• If the sample mean is within 10 points of µ, then
  µ is within 10 points of the sample mean.

• Therefore, the interval from 10 points below to
  10 points above the sample mean will capture µ
  in about 95 of all samples of size 16.

If we estimate that µ lies somewhere in the
interval 230.79 to 250.79, wed be calculating an
interval using a method that captures the true µ
in about 95 of all possible samples of this size.
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Confidence Interval
estimate margin of error

• Confidence Interval
• A level C confidence interval for a parameter has
  two parts
• An interval calculated from the data, which has
  the form
• estimate margin of error
• A confidence level C, which gives the
  probability that the interval will capture the
  true parameter value in repeated samples. That
  is, the confidence level is the success rate for
  the method.

We usually choose a confidence level of 90 or
higher because we want to be quite sure of our
conclusions. The most common confidence level is
95.
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Confidence Interval
The confidence level is the overall capture rate
if the method is used many times. The sample mean
will vary from sample to sample, but when we use
the method estimate margin of error to get an
interval based on each sample, C of these
intervals capture the unknown population mean µ.
Interpreting a Confidence Level
To say that we are 95 confident is shorthand for
95 of all possible samples of a given size from
this population will result in an interval that
captures the unknown parameter.
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Confidence Intervals for a Population Mean
Previously, we estimated the mystery mean µ by
constructing a confidence interval using the
sample mean 240.79.
To calculate a 95 confidence interval for µ , we
use the familiar formula estimate (critical
value) (standard deviation of statistic)
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Confidence Intervals The Four-Step Process
Confidence Intervals The Four-Step Process
State What is the practical question that
requires estimating a parameter? Plan Identify
the parameter, choose a level of confidence, and
select the type of confidence interval that fits
your situation. Solve Carry out the work in two
phases 1. Check the conditions for the interval
that you plan to use. 2. Calculate the
confidence interval. Conclude Return to the
practical question to describe your results in
this setting.
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How Confidence Intervals Behave

• The z confidence interval for the mean of a
  Normal population illustrates several important
  properties that are shared by all confidence
  intervals in common use.
• The user chooses the confidence level and the
  margin of error follows.
• We would like high confidence and a small margin
  of error.
• High confidence suggests our method almost always
  gives correct answers.
• A small margin of error suggests we have pinned
  down the parameter precisely.

How do we get a small margin of error?

• The margin of error for the z confidence interval
  is
• The margin of error gets smaller when
• z gets smaller (the same as a lower confidence
  level C)
• s is smaller. It is easier to pin down µ when s
  is smaller.
• n gets larger. Since n is under the square root
  sign, we must take four times as many
  observations to cut the margin of error in half.

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Chapter 14 Objectives Review

• Define statistical inference
• Describe the reasoning of statistical estimation
• Describe the parts of a confidence interval
• Interpret a confidence level
• Construct and interpret a confidence interval for
  the mean of a Normal population
• Describe how confidence intervals behave